Error Resilience of Fracton Codes and Near Saturation of Code-Capacity Threshold in Three Dimensions

TL;DR

This paper evaluates the fault-tolerance thresholds of three-dimensional fracton codes, especially the checkerboard code, showing they nearly reach the theoretical maximum capacity for topological quantum error correction.

Contribution

It provides the first comprehensive threshold estimates for self-dual fracton codes using statistical mechanics and Monte Carlo simulations, demonstrating their high resilience and near-saturation of capacity limits.

Findings

Checkerboard code threshold p_th ≈ 0.107(3)

Threshold nearly saturates the theoretical limit for topological codes

Validation of entropy duality relation in fracton codes

Abstract

Fracton codes have been intensively studied as novel topological states of matter, yet their fault-tolerant properties remain largely unexplored. Here, we investigate the optimal thresholds of self-dual fracton codes, in particular the checkerboard code, against stochastic Pauli noise. By utilizing a statistical-mechanical mapping combined with large-scale parallel tempering Monte Carlo simulations, we calculate the optimal code capacity of the checkerboard code to be $p_{th} \simeq 0.107(3)$. This value is the highest among known three-dimensional codes and nearly saturates the theoretical limit for topological codes. Our results further validate the generalized entropy relation for two mutually dual models, $H(p_{th}) + H(\tilde{p}_{th}) \approx 1$, and extend its applicability beyond standard topological codes. This verification indicates the Haah's code also possesses a code capacity near the theoretical limit $p_{th} \approx 0.11$. These findings highlight fracton codes as highly resilient quantum memory and demonstrate the utility of duality techniques in analyzing intricate quantum error-correcting codes.